Point equation of the boundary of the numerical range of a matrix polynomial
نویسندگان
چکیده
The numerical range of an n× n matrix polynomial P (λ) = Amλ m + Am−1λ m−1 + . . . + A1λ + A0 is defined by W (P ) = {λ ∈ C : x∗P (λ)x = 0, x ∈ C, x 6= 0}. For the linear pencil P (λ) = Iλ− A, the range W (P ) coincides with the numerical range of matrix A, F (A) = {x∗Ax : x ∈ C, x∗x = 1}. In this paper, we obtain necessary conditions for the origin to be a boundary point of F (A). As a consequence, an algebraic curve of degree at most 2n(n− 1)m, which contains the boundary of W (P ), is constructed.
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